Origins of the Capstan EquationA rope under Tension lying on a curved surface applies a force per length normal to that surface. Starting at the loaded end for the sake of discussion, this Normal Force in each small section of the rope is proportional to the Tension at that point and results in a Frictional Force which helps to support the load and reduces the Tension for the next small section of rope. In this next small section of rope, the Tension and therefore Normal Force AND Frictional Force are a little less than the previous section. Thus the change (decrease) in Tension is most rapid starting at the loaded end and this decrease continuously slows down as we progress to the held end of the rope.

Some definitions:

*T* :
| **Tension in the Rope** |

*N* :
| **Normal Force per Length Applied to the Cylinder Surface** |

*F*_{f} :
| **Frictional Force Applied to the Rope** |

**Δ :**
| **Signifies a Small Change in or Increment of a Variable** |

*s* :
| **A Length of the Curved Rope ( Arc )** |

*k* :
| **The Curvature of the Rope** |

The Frictional Force from a small length (section) of rope, Δ

*s*, can be calculated by multiplying the Normal Force per Length,

*N*, times this length Δ

*s*, times the Coefficient of Friction,

*μ*. In symbols:

*F*_{f} =

*N**Δ

*s**

*μ* Since this Frictional Force from this small section of rope is equivalent to the small decease in the Tension, Δ

*T*, the above can be rewritten as

Δ

*T* =

*μ* *

*N**Δ

*s* ( terms on the right side of the equation were also rearranged )

As mentioned previously, it is the Tension in the rope which results in this Normal Force per Length. The constant of proportionality is just the Curvature of the Rope,

*k*, lying on this cylindrical surface. The more tightly curved the surface, the larger the Curvature and the larger the Normal Force for the same applied Tension in the rope. In symbols:

*N* =

*k**

*T*Putting the last two equations together, eliminating N, produces after rearranging

Δ

*T* =

*μ* *

*k**Δ

*s**

*T* Dividing both sides by T results in

Δ

*T* /

*T* = [

*μ* *

*k* ]*Δ

*s* The form of this equation, the Change of a variable divided by that same variable, in this case

*T*, and proportional to a Change in a second variable, in this case

*s*, describes an exponential relationship with the argument of that exponential being the proportionality constant, in this case [

*μ* *

*k* ], times that second variable,

*s*. In symbols,

*T* =

*T*_{0}*

e^{ μks} This is the Generalized Form of the Capstan Equation for constant curvature surfaces.

*T*_{0} represents the tension at any starting point and

*s* represents the arc length from that same starting point to the point of interest which will have Tension

*T*.

Another example of such a relationship is population. The increase in the population is proportional to the current population times an increment of time, Δ

*t*. Thus population,

*P*, follows an exponential relationship.

*P* =

*P*_{0}*

e^{ βt} where β is a proportionality constant from

Δ

*P* = [ β ]*Δ

*t**

*P* or rewritten as Δ

*P* /

*P* = [ β ]*Δ

*t*